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Stewart
A transformation \(\mathbf{R}^2 \to \mathbf{R}^2\) is defined by the equations \({x=u^2-v^2}\) and \({y=2uv.}\) Sketch a picture of the image of the unit square \({(u,v) \in [0,1]\times[0,1]}\) under this transformation. -
Stewart
Use the change of coordinates \({x=u^2-v^2}\) and \({y=2uv}\) to evaluate the integral \(\iint_R y \,\mathrm{d}A\) over the region \(R\) bounded by the \(x\)-axis and the parabolas \(y^2=4-4x\) and \(y^2=4+4x\) for \(y \geq 0.\) -
Stewart
Evaluate the following integral over the trapezoidal region \(R\) with vertices \((1,0)\) and \((2,0)\) and \((0,-2)\) and \((0,-1).\)\(\displaystyle \iint\limits_R \mathrm{e}^{(x+y)/(x-y)}\,\mathrm{d}A \) -
Stewart
Consider the following change of coordinates \((\rho, \theta, \phi) \to (x,y,z)\) and calculate the corresponding Jacobian.\(\displaystyle x=\rho\sin(\phi)\cos(\theta)\)\(\displaystyle y=\rho\sin(\phi)\sin(\theta)\)\(\displaystyle z=\rho\cos(\phi)\)